Theory of Parabolas

A parabola is designed on a basic formula, Y=ax^2+bx+c, which allows it to achieve a curve not seen in a normal line graphed using a Y=mx+b format. To the left is a graph who’s formula is y=x^2, where a=1, b=0, and c=0. I have isolated the (a) factor to see its effects on the parabola. Below is a graph where I have changed the (a) multiple times. The result is that as the (a) decreases, the mouth of the parabola widens due to the fact that (a) is essentially the slope of the parabola.

Now, watch what happens when the (a) becomes negative. Now the mouth of the parabola is opening down and as we increase (a), the parabola widens. When (a)1, the parabola closes up horizontally. So from this, we can infer that if (a) is 0, then the result will be a straight line going along the X axis which is no longer a parabola. Now we will look at how the (b) factor changes the parabola, below is the origional graph from the top of the paper except the red line represents the same parabola with a (b) added onto it.

The original equation was Y=x^2(blue line) and the new equation is Y=x^2+x(red line). The change is moving the vertex of the parabola left ? and down ? but the shape of the parabola it’s self is unchanged. Here are some other examples of changing (b) in a parabola with a steady (a) As seen in the graph, (b) only moves the parabola to the sides and down (as the numbers increase, the vertex moves left and down and decreasing is vice versa), but somehow fails to move the parabola up past the point X=0 on its own (making the (a) negative will flip the parabola).

So now we have a slope (a) and a vertex (b), but we still need our Y-intercept which will be our (c) As shown from the graph, our parabolas Y-intercepts match up with our (c) factor, completing the equation -Fun fact, the word parabola came from the Greek words para, meaning beside, and bole, meaning to throw